SSC CGL Tier 1 2021 Quantitative Aptitude Questions with Solutions

Explanation

The train travels from station A to B between 8 am and 12 noon, taking 4 hours. The car travels from station B to A between 8:30 am and 12 noon, taking 3.5 hours or 7/2 hours. Let the distance between A and B be D.

The speed of the train is v_T = D/4 and the speed of the car is v_C = D/(7/2) = 2D/7.

Let the time they meet be t hours after 8 am. By this time, the train has traveled for t hours and the car has traveled for (t - 0.5) hours, since it started at 8:30 am. The sum of the distances covered by the train and the car at the meeting point equals the total distance D:

v_T · t + v_C · (t - 0.5) = D

Substituting the speeds: (D/4)t + (2D/7)(t - 0.5) = D

Dividing the entire equation by D: t/4 + (2/7)(t - 0.5) = 1 t/4 + 2t/7 - 1/7 = 1 (7t + 8t)/28 = 1 + 1/7 15t/28 = 8/7 t = (8/7) · (28/15) = 32/15 hours

Converting 32/15 hours into hours and minutes: 32/15 = 2 hours and 2/15 hours 2/15 hours = (2/15) · 60 minutes = 8 minutes

The meeting time is 2 hours and 8 minutes after 8 am, which is 10:08 am.

Explanation

In △ ABF, since DE ∥ AF, we apply the Basic Proportionality Theorem to get (BD)/(DA) = (BE)/(EF). Let EF be x. Substituting the given value BE = 4, we have (BD)/(DA) = ⁴⁄ₓ.

In △ ABC, since DF ∥ AC, the Basic Proportionality Theorem gives (BD)/(DA) = (BF)/(FC). The segment BF is the sum of BE and EF, so BF = 4 + x. Given CF = 3, we have (BD)/(DA) = (4 + x)/(3).

Equating the two expressions for (BD)/(DA): ⁴⁄ₓ = (4 + x)/(3) 12 = x(4 + x) x² + 4x - 12 = 0 (x + 6)(x - 2) = 0

Since the length x must be a positive value, x = 2. Thus, the length of EF is 2 cm.

Explanation

Let the total work be the Least Common Multiple of the time taken by A and B, which is LCM(15, 10) = 30 units.

The efficiency of A is 30 / 15 = 2 units per day, and the efficiency of B is 30 / 10 = 3 units per day. Their combined efficiency is 2 + 3 = 5 units per day. In the first 4 days, they complete 5 × 4 = 20 units of work.

The remaining work is 30 - 20 = 10 units. This remaining work is completed by A and C together in 3 days. Therefore, their combined efficiency is 10 / 3 units per day. Subtracting A's efficiency gives the efficiency of C: E_C = ¹⁰⁄₃ - 2 = ⁴⁄₃ units per day.

The question asks for the time taken by C to complete 40% of the total work. Target work = 40% of 30 = 0.4 × 30 = 12 units. Time taken by C = (Target work)/(Efficiency of C) = (12)/(4/3) = (12 × 3)/(4) = 9 days.

Explanation

The expression can be simplified by converting all trigonometric ratios into sin θ and cos θ.

First, simplify the term (cot³ θ)/(csc² θ): (cot³ θ)/(csc² θ) = (cos³ θ)/(sin³ θ) · sin² θ = (cos³ θ)/(sin θ)

Next, simplify the term (tan³ θ)/(sec² θ): (tan³ θ)/(sec² θ) = (sin³ θ)/(cos³ θ) · cos² θ = (sin³ θ)/(cos θ)

Substitute these back into the original expression: (cos³ θ)/(sin θ) + (sin³ θ)/(cos θ) + 2sin θ cos θ

Taking the common denominator sin θ cos θ for all terms: (cos⁴ θ + sin⁴ θ + 2sin² θ cos² θ)/(sin θ cos θ)

The numerator is in the form a² + b² + 2ab, where a = sin² θ and b = cos² θ: ((sin² θ + cos² θ)²)/(sin θ cos θ)

Using the fundamental identity sin² θ + cos² θ = 1: (1²)/(sin θ cos θ) = (1)/(sin θ) · (1)/(cos θ)

Applying reciprocal identities (1)/(sin θ) = csc θ and (1)/(cos θ) = sec θ: csc θ sec θ

Explanation

A number is divisible by 88 if it is simultaneously divisible by 8 and 11, as 8 and 11 are coprime.

For the number 7p5964q28 to be divisible by 8, its last three digits q28 must be divisible by 8. Checking the possible single-digit values for q, we find that 128, 328, 528, 728, and 928 are all divisible by 8. Since the question specifies the largest value of q, we take q = 9.

Now, for the number 7p5964928 to be divisible by 11, the difference between the sum of the digits at odd positions and the sum of the digits at even positions must be 0 or a multiple of 11. Sum of digits at odd positions (from right): 8 + 9 + 6 + 5 + 7 = 35 Sum of digits at even positions (from right): 2 + 4 + 9 + p = 15 + p Difference: 35 - (15 + p) = 20 - p

For 20 - p to be a multiple of 11 where p is a single-digit natural number, the only possible value is 20 - p = 11, which gives p = 9.

Substituting the values p = 9 and q = 9 into the required expression: p² - q = 9² - 9 = 81 - 9 = 72

Explanation

To find the year with the maximum number of trees planted, we calculate the sum of trees across all four cities for each year provided in the table. The total for 2016 is 1800 + 2500 + 1800 + 2000 = 8100. The total for 2017 is 2500 + 2300 + 1850 + 1800 = 8450. The total for 2018 is 2300 + 2400 + 1840 + 1760 = 8300. The total for 2019 is 2440 + 1950 + 1900 + 1600 = 7890. The total for 2020 is 2250 + 2100 + 2000 + 1750 = 8100. Comparing these annual sums, the value 8450 is the highest, which corresponds to the year 2017.

Explanation

Arshad received 35% of the total votes. Chamani received 14 votes for every 35 votes, which as a percentage is ¹⁴⁄₃₅ × 100 = ⅖ × 100 = 40%. Since all votes polled were valid and shared among three contestants, the percentage of votes received by Jagjit Singh is 100% - (35% + 40%) = 25%.

Comparing the three contestants, Chamani is the winner with 40% and Jagjit Singh received the least number of votes with 25%. The difference between their shares is 40% - 25% = 15%. Given that this difference equals 4950 votes, the total number of votes V can be calculated using the relation 15% of V = 4950.

Solving for V: V = (4950 × 100)/(15) V = 330 × 100 = 33000

Explanation

The angle ∠ ACD and ∠ ABD are both subtended by the same arc AD at the circumference of the circle. Therefore, ∠ ABD = ∠ ACD = 25°. Since AB is a diameter, the angle subtended by it at any point on the circumference is a right angle, which means ∠ ADB = 90°.

In triangle △ ABD, the sum of interior angles is 180°. Thus, we can calculate ∠ DAB as follows: ∠ DAB = 180° - (∠ ADB + ∠ ABD) = 180° - (90° + 25°) = 65°

The points A, D, E, and B lie on the circle. Given that E is on the minor arc BD, the vertices A, D, E, and B form a cyclic quadrilateral in that order. In any cyclic quadrilateral, the sum of opposite angles is 180°. The angle ∠ DAB and the angle ∠ BED are opposite angles in the cyclic quadrilateral ADEB. Consequently: ∠ BED = 180° - ∠ DAB = 180° - 65° = 115°

Explanation

For two successive discounts of d₁% and d₂%, the single equivalent discount is calculated using the formula D = d₁ + d₂ - d₁ × d₂{100}. In this case, the sum of the discounts for A, B, and C is the same: 20 + 20 = 40, 25 + 15 = 40, and 30 + 10 = 40. Since the sum is constant, the maximum discount occurs when the product d₁ × d₂ is the smallest.

Comparing the products: for A, 20 × 20 = 400; for B, 25 × 15 = 375; and for C, 30 × 10 = 300. Shopkeeper C gives the maximum discount because 300 is the smallest product among the three. The equivalent discount percentage for C is 30 + 10 - ³⁰⁰⁄₁₀₀ = 37%. The discount amount is 37% of the marked price: 4820 × ³⁷⁄₁₀₀ = 48.2 × 37 = 1783.40.

Explanation

The total number of coins is given as 72, distributed in the ratio 10 : 5 : 2 : 1 for denominations of ₹1, ₹2, ₹5, and ₹10 respectively. Let the common multiplier for the number of coins be x. The sum of the ratios is 10 + 5 + 2 + 1 = 18. Since the total number of coins is 72, we have 18x = 72, which gives x = 4.

The total value of the money in the piggy bank is the sum of the products of each denomination and its corresponding count. The value V is calculated as: V = (1 × 10x) + (2 × 5x) + (5 × 2x) + (10 × 1x) V = 10x + 10x + 10x + 10x = 40x

Substituting the value of x = 4 into the equation: V = 40 × 4 = 160

Each denomination contributes exactly ₹40 to the total, resulting in a total amount of ₹160.

Explanation

In simple interest, the interest earned remains constant every year. The difference between the amounts after 5 years and 3 years represents the simple interest for 2 years.

Simple Interest for 2 years = 16800 - 14880 = 1920 Simple Interest for 1 year = ¹⁹²⁰⁄₂ = 960 Simple Interest for 3 years = 960 × 3 = 2880

The principal (P) can be calculated by subtracting the interest for 3 years from the amount after 3 years: P = 14880 - 2880 = 12000

Now, we need to find the simple interest on the same sum at 10% per annum for 4 years. Using the formula SI = (P × R × T)/(100): SI = (12000 × 10 × 4)/(100) = 120 × 40 = 4800

Explanation

According to the property of tangents drawn from an external point to a circle, the lengths of the tangents are equal. From point A, the tangents AB and AC are equal, so AB = AC = 11 cm. Similarly, from point P, the segments PB and PR are tangents to the circle, implying PB = PR. From point Q, the segments QC and QR are tangents, implying QC = QR.

The length of PB is calculated as AB - AP = 11 - 7 = 4 cm. Therefore, PR = 4 cm. The length of QC is calculated as AC - AQ = 11 - 9 = 2 cm. Therefore, QR = 2 cm. The total length of the segment PQ is the sum of the segments PR and RQ, which gives PQ = PR + QR = 4 + 2 = 6 cm.

Explanation

The canvas required to cover a conical heap of wheat is equal to the curved surface area (CSA) of the cone. Given the radius r = 4.2 m and height h = 1.75 m, the slant height l is calculated as √(4.2² + 1.75²) = √20.7025 = 4.55 m. The area of the canvas is πrl = (22/7) × 4.2 × 4.55 = 60.06 m².

Explanation

Given the equation 2sin (3x - 15)° = 1, we simplify it to sin (3x - 15)° = 1/2. Since sin 30° = 1/2 and the problem specifies the angle is in the first quadrant, we set 3x - 15 = 30. Solving for x, we get 3x = 45, so x = 15.

Now, substitute x = 15 into the expression cos² (2x + 15)° + cot² (x + 15)°: cos² (2 × 15 + 15)° + cot² (15 + 15)° = cos² 45° + cot² 30°

Using standard trigonometric values where cos 45° = 1/√2 and cot 30° = √3, the calculation becomes: (1/√2)² + (√3)² = 1/2 + 3 = 7/2

Explanation

The total cost price of the transaction is 10000 + 5000 = 15000. To achieve a 10% gain on the whole transaction, the target total profit is 0.10 × 15000 = 1500.

The profit earned from selling the computer table at 8% is 0.08 × 10000 = 800. To meet the overall target, the profit needed from the centre table is 1500 - 800 = 700. The required profit percentage for the centre table is calculated as ⁷⁰⁰⁄₅₀₀₀ × 100 = 14%.

This can also be solved using the weighted average method. The ratio of the cost prices of the computer table to the centre table is 10000:5000, which simplifies to 2:1. Let x be the required profit percentage for the centre table. The equation for the overall profit is: (2 × 8 + 1 × x)/(2 + 1) = 10 16 + x = 30 x = 14%

Explanation

In △ ABC, since DE ∥ AB where D and E are points on sides AC and BC respectively, △ CDE is similar to △ CAB by the AA similarity criterion. According to the Basic Proportionality Theorem and the properties of similar triangles, the segments of the sides are proportional. Specifically, the ratio of the segment from the parallel line to the vertex on one side to the total length of that side must equal the corresponding ratio on the other side:

(AD)/(AC) = (BE)/(BC)

Substituting the given expressions into this proportionality relation:

(x - 3)/(2x) = (x - 2)/(2x + 3)

To find the value of x, cross-multiply the terms:

(x - 3)(2x + 3) = 2x(x - 2)

Expanding both sides of the equation:

2x² + 3x - 6x - 9 = 2x² - 4x 2x² - 3x - 9 = 2x² - 4x

Subtracting 2x² from both sides and simplifying:

-3x - 9 = -4x 4x - 3x = 9 x = 9

Explanation

In compound interest, the amount at the end of a year acts as the principal for the following year. Therefore, the interest earned in the third year is the difference between the amounts at the end of the third year and the second year.

Interest for the 3rd year = 314928 - 291600 = 23328

The annual interest rate r is calculated as: r = A₃ - A₂{A₂} × 100 = ²³³²⁸⁄₂₉₁₆₀₀ × 100 = 8%

The simple interest (SI) on a principal of ₹40000 for 2 years at the same rate of 8% is given by the formula: SI = (P × r × t)/(100) SI = (40000 × 8 × 2)/(100) = 400 × 16 = 6400

Explanation

Given the trigonometric equations sin ((2A + B)/2) = √3/2 and cos ((2A - B)/2) = √3/2, we identify the corresponding angles in the first quadrant. Since sin 60° = √3/2 and cos 30° = √3/2, we set up the following equations:

(2A + B)/2 = 60° ⟹ 2A + B = 120°

(2A - B)/2 = 30° ⟹ 2A - B = 60°

Adding these two equations eliminates B: (2A + B) + (2A - B) = 120° + 60° 4A = 180° ⟹ A = 45°

Subtracting the second equation from the first eliminates A: (2A + B) - (2A - B) = 120° - 60° 2B = 60° ⟹ B = 30°

Now, calculate the value of A - B: A - B = 45° - 30° = 15°

Substitute this into the expression sin [3(A - B)]: sin [3(15°)] = sin 45° = 1/√2

Explanation

The relationship between Cost Price (CP) and Marked Price (MP) is given by the formula (CP)/(MP) = (100 - Discount%)/(100 + Profit%). Substituting the initial values of 16% discount and 8% profit, we get:

(CP)/(MP) = (100 - 16)/(100 + 8) = ⁸⁴⁄₁₀₈ = ⁷⁄₉

In the second case, the discount is changed to 8%. Let the new profit be x%. Using the same ratio:

⁷⁄₉ = (100 - 8)/(100 + x)

⁷⁄₉ = (92)/(100 + x)

700 + 7x = 828

7x = 128

x = ¹²⁸⁄₇ ≈ 18.2857

Correct to two decimal places, the profit percent is 18.29%.

Explanation

Using the VBODMAS rule, we first evaluate the terms within parentheses and the of operations.

In the first part of the expression, 15 of 4 = 15 × 4 = 60. Inside the square bracket, the inner parentheses simplify to 9 - 7 = 2, making the bracket [4 ÷ 5 × 2] = ⅘ × 2 = ⁸⁄₅. In the second part of the expression, the term 5 of 9 = 5 × 9 = 45, so the parenthetical term (20 ÷ 5 of 9) becomes 20 ÷ 45 = ²⁰⁄₄₅ = ⁴⁄₉.

Now, substituting these simplified values back into the original expression gives: 25 ÷ 60 × ⁸⁄₅ - ⁴⁄₉

Next, we perform the division and then the multiplication: ²⁵⁄₆₀ × ⁸⁄₅ - ⁴⁄₉ ⁵⁄₁₂ × ⁸⁄₅ - ⁴⁄₉ ⁸⁄₁₂ - ⁴⁄₉ ⅔ - ⁴⁄₉

To complete the subtraction, we use a common denominator of 9: ⁶⁄₉ - ⁴⁄₉ = ²⁄₉

Explanation

The expression p³ - 8q³ can be viewed as p³ - (2q)³. Using the algebraic identity for the difference of cubes, a³ - b³ = (a - b)³ + 3ab(a - b), we set a = p and b = 2q.

Expanding the identity with these values: p³ - (2q)³ = (p - 2q)³ + 3(p)(2q)(p - 2q) p³ - 8q³ = (p - 2q)³ + 6pq(p - 2q)

Substitute the given values p - 2q = 3 and pq = 5 into the equation: p³ - 8q³ = (3)³ + 6(5)(3) p³ - 8q³ = 27 + 90 p³ - 8q³ = 117

Explanation

Let the upstream speed be u km/h and the downstream speed be v km/h. According to the first condition, 27/u + 33/v = 6. For the second condition, 36/u + 22/v = 6. Equating these two expressions since they both equal 6 gives:

27/u + 33/v = 36/u + 22/v

Subtracting 27/u from 36/u and 22/v from 33/v, we get 11/v = 9/u, which implies u = ⁹⁄₁₁v.

Substitute u into the second equation: 36/(9v/11) + 22/v = 6 44/v + 22/v = 6 66/v = 6 v = 11 km/h

Substituting v = 11 into u = ⁹⁄₁₁v gives u = 9 km/h. To find the time taken for 36 km upstream and 44 km downstream, we use the time formula T = Dᵤ/u + D_d/v:

T = 36/9 + 44/11 T = 4 + 4 = 8 hours

Explanation

Let r be the radius of the circle and O be the centre. The perpendicular from the centre to a chord bisects the chord. For chord CD = 18 cm, the half-length is 9 cm. For chord AB = 12 cm, the half-length is 6 cm. Let x be the perpendicular distance from O to chord CD. Since AB is shorter than CD and both are on the same side of the diameter, AB is farther from the centre. The distance between them is 3 cm, so the perpendicular distance from O to AB is x + 3.

Using the Pythagorean theorem in the two right-angled triangles formed by the radius, the half-chords, and the perpendicular distances from the centre: r² = x² + 9² r² = (x + 3)² + 6²

Equating the two expressions for r²: x² + 81 = (x + 3)² + 36 x² + 81 = x² + 6x + 9 + 36 81 = 6x + 45 6x = 36 x = 6

Substitute x = 6 into the first equation: r² = 6² + 9² r² = 36 + 81 r² = 117 r = √(117) = √(9 × 13) = 3√(13) cm

Explanation

To find the value of x³ - 8y³, we use the algebraic identity a³ - b³ = (a - b)(a² + ab + b²). In this case, let a = x and b = 2y, so the expression becomes: x³ - (2y)³ = (x - 2y)(x² + 2xy + 4y²)

We are given x - 2y = 5 and x² + 4y² = 53. To find the term 2xy, we square the first given equation: (x - 2y)² = 5² x² - 4xy + 4y² = 25 Substitute x² + 4y² = 53 into this equation: 53 - 4xy = 25 4xy = 53 - 25 = 28 xy = 7

Now, calculate the required value of 2xy: 2xy = 2(7) = 14

Substitute all the values back into the factored form of the cubic expression: x³ - 8y³ = (x - 2y)(x² + 4y² + 2xy) x³ - 8y³ = 5(53 + 14) x³ - 8y³ = 5(67) x³ - 8y³ = 335

Explanation

Substitute the standard trigonometric values into the given expression. We know that sin 30° = ½, cos 30° = √3{2}, sin 45° = (1)/(√(2)), and cos 45° = (1)/(√(2)).

The first part of the expression is 4(sin⁴30° + cos⁴30°). Substituting the values, we get: 4((½)⁴ + (√3{2})⁴) = 4(¹⁄₁₆ + ⁹⁄₁₆) = 4(¹⁰⁄₁₆) = ¹⁰⁄₄ = ⁵⁄₂.

The second part of the expression is 3(sin²45° - 2cos²45°). Substituting the values, we get: 3(((1)/(√(2)))² - 2((1)/(√(2)))²) = 3(½ - 2(½)) = 3(½ - 1) = 3(-½) = -³⁄₂.

Subtracting the second part from the first: ⁵⁄₂ - (-³⁄₂) = ⁵⁄₂ + ³⁄₂ = ⁸⁄₂ = 4.

Explanation

Let the total work be the least common multiple of 10 and 15, which is 30 units. Based on this, the efficiency of Samir is 30 / 10 = 3 units per day, and the efficiency of Puneet is 30 / 15 = 2 units per day.

The total duration of the work was 5 days. Since Samir and Puneet worked from the beginning until the completion, they each worked for the entire 5 days. The work completed by Samir in 5 days is 3 × 5 = 15 units. The work completed by Puneet in 5 days is 2 × 5 = 10 units. Together, Samir and Puneet completed 15 + 10 = 25 units of the 30-unit task.

The remaining work was completed by Ashok during the period he was involved. The work done by Ashok is 30 - 25 = 5 units. Since the total payment of ₹4500 is distributed in proportion to the amount of work done by each individual, Ashok's share is calculated as: Amount paid to Ashok = ((Work done by Ashok)/(Total work)) × Total payment Amount paid to Ashok = (⁵⁄₃₀) × 4500 = ⅙ × 4500 = 750

Explanation

In △ ABC, let us draw a line segment DG parallel to EC such that G lies on side AB.

In △ AEC, D is the midpoint of AC and DG ∥ EC. By the Midpoint Theorem (or the Converse of the Basic Proportionality Theorem), G must be the midpoint of AE. Therefore, the segment AE is divided into two equal parts AG and GE. AG = GE = (AE)/2 = 30/2 = 15 cm

In △ BDG, it is given that EC bisects BD at F, meaning F is the midpoint of BD. Since FE is a segment of the line EC and we constructed DG ∥ EC, it follows that FE ∥ DG. By the Midpoint Theorem applied to △ BDG, since F is the midpoint of BD and FE ∥ DG, E must be the midpoint of BG. Therefore, the segment BG is divided into two equal parts BE and EG. BE = EG

Since we already calculated GE = 15 cm, we find that: BE = 15 cm

Explanation

In a circle with center O, the angle subtended by an arc at the center is double the angle subtended by it at any point on the remaining part of the circle. Since ∠ ACB = 115° is an obtuse angle, the vertex C lies on the minor arc, and the angle ∠ ACB subtends the reflex angle at the center O.

The reflex angle ∠ AOB is calculated as: Reflex ∠ AOB = 2 × ∠ ACB = 2 × 115° = 230°

The interior angle ∠ AOB within △ OAB is: ∠ AOB = 360° - 230° = 130°

In △ OAB, the segments OA and OB are radii of the circle, meaning OA = OB. Therefore, △ OAB is an isosceles triangle, and the angles opposite to the equal sides are equal, so ∠ OAB = ∠ OBA.

Applying the angle sum property in △ OAB: ∠ OAB + ∠ OBA + ∠ AOB = 180° 2 × ∠ OAB + 130° = 180° 2 × ∠ OAB = 50° ∠ OAB = 25°

Explanation

Let the cost price of the article sold at a loss be CP₁ and the cost price of the article sold at a profit be CP₂. Since the selling prices are equal, we can equate the expressions for selling price based on their respective profit and loss percentages.

The first article is sold at a 20% loss, so SP₁ = 80% of CP₁. The second article is sold at a 16% profit, so SP₂ = 116% of CP₂. Setting SP₁ = SP₂:

0.80 × CP₁ = 1.16 × CP₂

CP₁{CP₂} = (1.16)/(0.80) = ¹¹⁶⁄₈₀ = ²⁹⁄₂₀

The ratio of the cost prices CP₁ : CP₂ is 29 : 20. The total cost price is given as ₹490. Using the ratio components, the total parts are 29 + 20 = 49.

The cost price of the article sold at a 20% loss (CP₁) is calculated as:

CP₁ = ²⁹⁄₄₉ × 490 = 29 × 10 = 290

Explanation

Let the initial number be x. When a number is increased by a% and then decreased by b%, the net effect is x × (1 + a/100) × (1 - b/100). Given the first set of operations: x × (1 + 15/100) × (1 - 15/100) = 782 x × (115/100) × (85/100) = 782 x × (23/20) × (17/20) = 782 x × (391/400) = 782 x = (782 × 400) / 391 x = 2 × 400 = 800

For the second part, the number 800 is reduced by 25%, increased by 25%, and then reduced by 20%. The resulting number is calculated as: Result = 800 × (1 - 25/100) × (1 + 25/100) × (1 - 20/100) Result = 800 × (3/4) × (5/4) × (4/5)

By observing the fractions, (5/4) × (4/5) = 1. The calculation simplifies to: Result = 800 × (3/4) × 1 Result = 200 × 3 = 600

Explanation

A number is divisible by 3 if the sum of its digits is a multiple of 3. For the number 5306P2, the sum of the digits is 5 + 3 + 0 + 6 + P + 2 = 16 + P.

To make 16 + P divisible by 3, the possible values for the single-digit P are: If 16 + P = 18, then P = 2. If 16 + P = 21, then P = 5. If 16 + P = 24, then P = 8.

The smallest value of P is 2 and the greatest value of P is 8. The difference between their squares is calculated as follows: 8² - 2² = 64 - 4 = 60

Explanation

The side of the square field is calculated as s = √(Area) = √(1764) = 42 m. According to the problem, the breadth b of the rectangular park is one-third the side of the square field, so b = ⅓ × 42 = 14 m. The length l of the park is twice its breadth, which gives l = 2 × 14 = 28 m.

The area of the rectangular park is l × b = 28 × 14 = 392 m². The total cost of levelling the park at a rate of ₹15 per m² is found by multiplying the area by the rate: 392 × 15 = 5880.

Explanation

The denominator follows the trigonometric identity sin A cos B - cos A sin B = sin(A - B). Substituting A = 60° and B = 30°, we get sin(60° - 30°) = sin 30° = 1/2.

For the numerator, we substitute the standard values: tan 30° = 1/√3, sin 90° = 1, cot 60° = 1/√3, sin 30° = 1/2, and cos 45° = 1/√2. The expression becomes: (1/√3)² + 1² + (1/√3)² + (1/2)² × (1/√2)² = 1/3 + 1 + 1/3 + (1/4 × 1/2) = 2/3 + 1 + 1/8 = 5/3 + 1/8 = (40 + 3) / 24 = 43/24

The value of the total expression is the ratio of the numerator to the denominator: (43/24)/(1/2) = ⁴³⁄₂₄ × 2 = ⁴³⁄₁₂

Explanation

In △ ABC, since DE ∥ BC, △ ADE is similar to △ ABC by the Angle-Angle (AA) similarity criterion. According to the property of similar triangles, the ratio of the areas of two similar triangles is equal to the square of the ratio of their corresponding sides.

Using the relation: (Area(△ ADE))/(Area(△ ABC)) = (DE/BC)²

Substituting the given values: (45)/(Area(△ ABC)) = (5/8)²

(45)/(Area(△ ABC)) = 25/64

Area(△ ABC) = (45 × 64)/25

Area(△ ABC) = (9 × 64)/5

Area(△ ABC) = 576/5 = 115.2 cm²

Explanation

The length of the tangent drawn from a vertex to the opposite excircle of a triangle is equal to the semi-perimeter of the triangle. In △ ABC, the circle touches the extended sides AB and AC at P and Q respectively, and touches side BC at some point R. According to the properties of tangents from an external point, AP = AQ, BP = BR, and CQ = CR.

The perimeter of △ ABC is the sum of its sides: Perimeter = AB + BC + CA Perimeter = AB + (BR + RC) + CA

Substituting BR = BP and RC = CQ into the equation: Perimeter = (AB + BP) + (AC + CQ) Perimeter = AP + AQ

Since AP = AQ, the perimeter is equal to 2 × AQ. Therefore, the length of AQ is the semi-perimeter s: AQ = (AB + BC + CA)/2 AQ = (5 + 6 + 7)/2 AQ = 18/2 = 9 cm.

Explanation

Using the work equivalence formula (M₁ × D₁ = M₂ × D₂), let M and W represent the daily work efficiency of a man and a woman respectively. The given conditions provide the equation: (2M + 7W) × 28 = (6M + 16W) × 11

Expanding the terms: 56M + 196W = 66M + 176W 10M = 20W 1M = 2W

This efficiency ratio shows that one man does the work of two women. Now, substitute W = M/2 into the first expression to calculate the total work in terms of man-days: Total Work = (2M + 7(M/2)) × 28 = (2M + 3.5M) × 28 = 5.5M × 28 = 154M

To find the number of days D required for 7 men to complete this work: 7M × D = 154M D = 154 / 7 = 22

Explanation

In triangle ABF, since DE is parallel to AF, applying the Basic Proportionality Theorem (Thales's Theorem) gives the ratio (BD)/(DA) = (BE)/(EF). Substituting the given values BE = 4 cm and EF = 6 cm, we find (BD)/(DA) = ⁴⁄₆ = ⅔.

In triangle ABC, since DF is parallel to AC, the same theorem states that (BD)/(DA) = (BF)/(FC). The length of BF is the sum of BE and EF, which is 4 + 6 = 10 cm. Substituting the known ratio (BD)/(DA) = ⅔ into the equation, we get ⅔ = (10)/(FC). Solving for FC, we have FC = (10 × 3)/(2) = 15 cm.

The total length of BC is calculated by summing the segments BE, EF, and FC. Thus, BC = 4 + 6 + 15 = 25 cm.

Explanation

The expression is evaluated using the BODMAS rule, which dictates the order of operations as Brackets, Of, Division, Multiplication, Addition, and Subtraction.

First, simplify the terms associated with the of operator and the brackets. For the of terms, 12 of 3 = 12 × 3 = 36 and 2 of 4 = 2 × 4 = 8. Within the brackets, the operation 15 - 12 = 3 is performed first, followed by the division, giving 3 ÷ 9 = ⅓. This simplifies the square bracket to [5 - ⅓] = ¹⁴⁄₃.

Next, apply the of operator following the bracketed result. Using the intended value for the term rendered as 7³, we evaluate ¹⁴⁄₃ of ³⁄₇ = ¹⁴⁄₃ × ³⁄₇ = 2.

Now, substitute these simplified components back into the main expression: 32 ÷ 36 × 2 + 4 - 8 ÷ 8

Perform the division and multiplication operations from left to right. The division 32 ÷ 36 simplifies to ⁸⁄₉, and multiplying this by 2 yields ¹⁶⁄₉. The term 8 ÷ 8 simplifies to 1.

Finally, complete the addition and subtraction: ¹⁶⁄₉ + 4 - 1 = ¹⁶⁄₉ + 3

To add these, use a common denominator: ¹⁶⁄₉ + ²⁷⁄₉ = ⁴³⁄₉ = 4 ⁷⁄₉

Explanation

To find the value of the expression, we substitute the standard trigonometric values for the given angles: cot 30° = √3 cos 30° = √3{2} csc 60° = (2)/(√(3)) tan 60° = √3

Substituting these values into the expression ¾cot²30° + cos²30° - 3csc²60° + tan²60°:

= ¾(√3)² + (√3{2})² - 3((2)/(√(3)))² + (√3)²

= ¾(3) + ¾ - 3(⁴⁄₃) + 3

= ⁹⁄₄ + ¾ - 4 + 3

= ¹²⁄₄ - 1

= 3 - 1

= 2

Explanation

For the simple interest case, the amount A is given by A = P(1 + ʳᵗ⁄₁₀₀). Substituting the given values, 13650 = P(1 + (15 × 2)/(100)), which simplifies to 13650 = P(1.3). Solving for the principal, P = (13650)/(1.3) = 10500.

For the compound interest case, the interest is compounded half-yearly for 1 year. The effective rate for each six-month period is R = (15%)/(2) = 7.5%, and the number of compounding periods is n = 2. The compound amount formula is A = P(1 + (R)/(100))ⁿ. Substituting the values, A = 10500(1 + (7.5)/(100))² = 10500(1.075)².

Calculating the squared term, 1.075² = 1.155625. Multiplying this by the principal, A = 10500 × 1.155625 = 12134.0625. Rounding to the nearest rupee, the amount is 12134.

Explanation

The measure of ∠ AQB is determined using the Alternate Segment Theorem. This theorem states that the angle between a tangent to a circle and a chord through the point of contact is equal to the angle subtended by that chord in the alternate segment.

In the given circle, PAX is the tangent at point A and PBY is the tangent at point B. For the chord AQ and the tangent PAX, the angle ∠ QAX is given as 59°. According to the Alternate Segment Theorem, the angle subtended by the chord AQ at point B on the circumference (which is ∠ ABQ) is equal to ∠ QAX. Thus, ∠ ABQ = 59°.

Similarly, for the chord BQ and the tangent PBY, the angle ∠ QBY is given as 72°. By the same theorem, the angle subtended by the chord BQ at point A on the circumference (which is ∠ BAQ) is equal to ∠ QBY. Thus, ∠ BAQ = 72°.

Considering triangle ABQ, the sum of the interior angles is 180°. Therefore, the measure of ∠ AQB is calculated as: ∠ AQB = 180° - (∠ BAQ + ∠ ABQ) ∠ AQB = 180° - (72° + 59°) ∠ AQB = 180° - 131° = 49°

Explanation

The selling price of the article is SP = 640. The loss is given as 15% of this selling price, so Loss = 640 × ¹⁵⁄₁₀₀ = 96. The cost price (CP) of the article is calculated by adding the loss to the selling price: CP = 640 + 96 = 736. To earn a profit of 15% on the cost price, the new selling price must be 115% of the CP. Therefore, the required selling price is 736 × ¹¹⁵⁄₁₀₀ = 736 × 1.15 = 846.40.

Explanation

Total number of students enrolled in institutes A, C, and E in 2015 is calculated by summing the values from the corresponding cells in the table: 150 (A) + 220 (C) + 200 (E) = 570

Total number of students enrolled in institutes B and D in 2017 is calculated similarly: 200 (B) + 180 (D) = 380

The ratio of the total students in A, C, and E in 2015 to the total students in B and D in 2017 is: Ratio = 570 : 380

To simplify the ratio, divide both sides by 10 to get 57 : 38. Both 57 and 38 are multiples of 19. Dividing by 19 yields: 57 / 19 = 3 38 / 19 = 2

The simplified ratio is 3 : 2.

Explanation

The selling price (SP) is obtained by subtracting the discount amount from the marked price (MP). Given a marked price of 1500 and a discount rate of 20%, the selling price is calculated as: SP = 1500 × (1 - ²⁰⁄₁₀₀) = 1500 × 0.8 = 1200 Profit is defined as the difference between the selling price and the cost price (CP). With the cost price given as 991, the profit is determined by the formula: Profit = SP - CP = 1200 - 991 = 209

Explanation

The expression can be simplified by evaluating the two factors separately. First, consider the term inside the square root: sec²θ + csc²θ = (1)/(cos²⁾^θ + (1)/(sin²⁾^θ = sin²θ + cos²θ{sin²θ cos²θ} = (1)/(sin²⁾^θᶜᵒˢ^^^²^θ Taking the square root yields (1)/(sinθ cosθ).

Next, simplify the second factor. Using the identity 1 - sin²θ = cos²θ, the denominator becomes: 1 + cosθ - sin²θ = (1 - sin²θ) + cosθ = cos²θ + cosθ = cosθ(1 + cosθ) The second factor then simplifies as follows: (sinθ(1 + cosθ))/(cosθ(1 + cosθ)) = (sinθ)/(cosθ)

Multiplying the two simplified factors together: ( (1)/(sinθ cosθ) ) ( (sinθ)/(cosθ) ) = (1)/(cos²⁾^θ = sec²θ

Explanation

The algebraic identity relating the given terms is x³ + y³ + z³ - 3xyz = (x + y + z)(x² + y² + z² - (xy + yz + zx)). First, find the value of xy + yz + zx using the square of the sum formula (x + y + z)² = x² + y² + z² + 2(xy + yz + zx). Substituting the given values 7² = 85 + 2(xy + yz + zx) leads to 49 = 85 + 2(xy + yz + zx), which simplifies to 2(xy + yz + zx) = -36, so xy + yz + zx = -18.

Next, substitute these values into the cubic identity: 913 - 3xyz = 7(85 - (-18)). This results in 913 - 3xyz = 7(103), which gives 913 - 3xyz = 721. Rearranging to solve for xyz, we get 3xyz = 913 - 721 = 192, so xyz = 64. The required value is the cube root of this result: ³√(xyz) = ³√(64) = 4.

Explanation

A number is divisible by 72 if it is divisible by both 8 and 9. For divisibility by 8, the last three digits 9y6 must be divisible by 8. Testing values for y, we find that 936 (y=3) and 976 (y=7) are multiples of 8.

For divisibility by 9, the sum of the digits 9+4+x+2+9+y+6 = 30+x+y must be a multiple of 9. This implies 3+x+y is divisible by 9.

If we take y=3, then 3+x+3 = 6+x, which gives x=3 to make the sum 9. This is rejected because the problem states x ≠ y.

If we take y=7, then 3+x+7 = 10+x. To make this a multiple of 9, x must be 8 (since 10+8=18). This satisfies the condition x ≠ y.

Using x=8 and y=7, the value of (2x + 3y) is calculated as: 2(8) + 3(7) = 16 + 21 = 37

Explanation

For a set of n consecutive even numbers, the average is the middle term. Since there are 8 numbers, the average of 17 lies exactly between the 4ᵗʰ and 5ᵗʰ terms. The even numbers flanking 17 are 16 and 18. Therefore, the sequence in ascending order is 10, 12, 14, 16, 18, 20, 22, and 24.

The last three numbers of this sequence are 20, 22, and 24. To find the average of these three numbers along with 36 and 53, we calculate the sum of these five values: 20 + 22 + 24 + 36 + 53 = 155. The average is the sum divided by the count of numbers, which is ¹⁵⁵⁄₅ = 31.

Explanation

The total number of students studying Computer Science in institutes A and B is found by adding the values for those specific cells in the table: 57 + 48 = 105.

Next, the total number of students studying Arts in institutes B and C is found by summing the corresponding values: 45 + 55 = 100.

To determine by what percent the first group is more than the second group, the percentage increase formula is applied:

Percentage More = (Total CS (A+B) - Total Arts (B+C))/(Total Arts (B+C)) × 100

Substituting the values into the formula:

Percentage More = (105 - 100)/(100) × 100

Percentage More = ⁵⁄₁₀₀ × 100 = 5%

Explanation

The given equation a² + c² + 17 = 2(a - 8b - 2b²) can be rearranged by bringing all terms to the left side: a² - 2a + 4b² + 16b + c² + 17 = 0

To solve for the variables, express the equation as a sum of perfect squares: (a² - 2a + 1) + (4b² + 16b + 16) + c² = 0 (a - 1)² + (2b + 4)² + c² = 0

For the sum of squares of real numbers to be zero, each individual squared term must independently be equal to zero: a - 1 = 0 ⟹ a = 1 2b + 4 = 0 ⟹ b = -2 c = 0

Substitute these values into the expression (a³ + b³ + c³): 1³ + (-2)³ + 0³ = 1 - 8 + 0 = -7